Animated Solution for Physics - Kinematics: Match List I with List II.

The Triangle Law in Action

Vectors are the language of physics. They don't just tell us how much of something we have; they tell us where it's going. When we add vectors, we aren't just adding numbers; we are combining directions. This problem is a beautiful exercise in translating algebraic vector equations into geometric realities using the Triangle Law of Vector Addition.

Let's embark on a journey to decode these equations one by one.

The core principle we will use is simple: isolate one vector on one side of the equation. This isolated vector becomes our resultant.

Let's look at the first equation from List I:

If we rearrange this by moving $\mathbf{A}$ and $\mathbf{B}$ to the right side, we get:

What does this mean physically? It means that if you walk along vector $\mathbf{A}$, and then from the end of $\mathbf{A}$, you walk along vector $\mathbf{B}$, your net displacement from the starting point is exactly vector $\mathbf{C}$.

According to the Triangle Law, to add $\mathbf{A}$ and $\mathbf{B}$, we place the tail of $\mathbf{B}$ at the head of $\mathbf{A}$. The resultant vector $\mathbf{C}$ is then drawn from the free tail of $\mathbf{A}$ to the free head of $\mathbf{B}$.

When we look at the given images, Image (iv) perfectly illustrates this. Vector $\mathbf{A}$ is horizontal, vector $\mathbf{B}$ is vertical, and $\mathbf{C}$ is the hypotenuse starting from $\mathbf{A}$'s tail and ending at $\mathbf{B}$'s head. Therefore, (A) matches with (iv).

We apply the exact same logic to the next two equations.

For equation (B): Here, $\mathbf{A}$ is the resultant. We need a diagram where $\mathbf{C}$ and $\mathbf{B}$ are added head-to-tail, and $\mathbf{A}$ closes the triangle from the start. Image (iii) shows exactly this: $\mathbf{C}$ is horizontal, $\mathbf{B}$ is vertical, and $\mathbf{A}$ is the resultant. Thus, (B) matches with (iii).

For equation (C): Now, $\mathbf{B}$ is the resultant of $\mathbf{A}$ and $\mathbf{C}$. Looking at Image (i), we see $\mathbf{A}$ is horizontal, $\mathbf{C}$ is vertical, and $\mathbf{B}$ connects the tail of $\mathbf{A}$ to the head of $\mathbf{C}$. So, (C) matches with (i).

The final equation presents a slightly different, yet incredibly important, scenario:

If we bring $\mathbf{C}$ to the left side, we get:

This equation tells a story of a journey that ends exactly where it began. If you walk along $\mathbf{A}$, then along $\mathbf{B}$, and finally along $\mathbf{C}$, your net displacement is zero.

Geometrically, this means the vectors form a closed polygon (in this case, a triangle) when placed head-to-tail in sequence. There is no resultant vector because the head of the final vector meets the tail of the first vector.

Image (ii) is the perfect visual representation of this closed loop. Vector $\mathbf{A}$ goes right, $\mathbf{B}$ goes up, and $\mathbf{C}$ comes back down to the starting point. Therefore, (D) matches with (ii).

By simply rearranging the algebra, we unlocked the geometry of the problem. This head-to-tail visualization is a fundamental tool that will serve you well in kinematics, forces, and electromagnetism!